Information Processing: Contracts versus Communication€¦ · Contracts versus communication A...
Transcript of Information Processing: Contracts versus Communication€¦ · Contracts versus communication A...
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Information Processing:Contracts versus Communication
Andreas Blume Inga Deimen Sean Inoue
U Arizona U Arizona Colgate U
October 2020
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Introduction Model Example Results Extensions Conclusions
Contracts versus communication
• A principal expects to receive private information
• The principal relies on an agent acting on that information
• Ideally, complete control in advance:actions pre-specified for all contingencies
• Difficult to exercise this degree of control
• Role for non-binding ad hoc communication (cheap talk)
Questions:
When to commit (to instructions) and when to communicate?
How do contract and cheap talk interact?
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Introduction Model Example Results Extensions Conclusions
Timeline of the gameStage 1: Principal writes contract
• Codifes language that makes conditions (sets of states) andactions (instructions) verifiable to third parties
• Commits principal to provide instructions (not state)
• Commits agent to follow instructions• Incompleteness: effort to make conditions verifiable
• Finitely many instructions and potential gaps
θ
x
0 1C1 C2 C3
x1
x2
x3
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Introduction Model Example Results Extensions Conclusions
Timeline of the gameStage 1: Principal writes contract
• Codifes language that makes conditions (sets of states) andactions (instructions) verifiable to third parties
• Commits principal to provide instructions (not state)
• Commits agent to follow instructions• Incompleteness: effort to make conditions verifiable
• Finitely many instructions and potential gaps
θ
x
0 1C1 C2 C3
x1
x2
x3
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Introduction Model Example Results Extensions Conclusions
Timeline of the game
Stage 2: State realizes and is privately observed by principal
Stage 3: Principal communicates with agent:
• Sends instruction (contract)
• Sends cheap talk recommendation (gap)
Stage 4: Agent takes action
• Follows instruction (contract)
• Chooses action (gap)
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Introduction Model Example Results Extensions Conclusions
Timeline of the game
θ
x , y
0 1C1 C2 C3
x1
x2
x3
θ1 θ2 θ3
y1y2
y3
y4
Stage 5: States in the contract become verifiable
• Contract establishes language that makes states in thecontract verifiable
• Informal communication cannot be verified
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Introduction Model Example Results Extensions Conclusions
Timeline of the game
θ
x , y
0 1C1 C2 C3
x1
x2
x3
θ1 θ2 θ3
y1y2
y3
y4
Stage 5: States in the contract become verifiable
• Contract establishes language that makes states in thecontract verifiable
• Informal communication cannot be verified
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Introduction Model Example Results Extensions Conclusions
Take-aways
Tradeoff: ex ante commitment versus ex post discretion
General preferences:
• Small disagreement: communication dominates
• Many clauses: contracting dominates
Uniform-quadratic:
• Contracts relax incentive constraints in communication
• Benefits from contract• direct: shifts control to principal• indirect: more actions in communication
more equalized communication intervals
Example:
• Contracts cover states with more conflict
• Contracts cover states that are more likely
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Introduction Model Example Results Extensions Conclusions
Related literature
• Crawford and Sobel (1982) (Cheap talk)
• Dye (1985), Battigalli & Maggi (2002) (Writing cost)
• Shavell (2006), Heller & Spiegler (2008), Schwartz & Watson(2013) (Contract interpretation)
• Aumann & Hart (2003), Krishna & Morgan (2004), Golosov,Skreta, Tsyvinski, & Wilson (2014) (Endogenouscommunication thresholds)
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Introduction Model Example Results Extensions Conclusions
Model
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Introduction Model Example Results Extensions Conclusions
Model
• Players:• Sender (Principal)• Receiver (Agent)
• Receiver’s action y ∈ R
• State θ ∈ [0, 1] with θ ∼ F , f (θ) > 0
• Payoffs: standard concave loss functions satisfying positivemixed-partial condition
• Sender US (y , θ, b)• Receiver UR (y , θ)• positive sender-bias b > 0
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Introduction Model Example Results Extensions Conclusions
Timing of the contract writing game G
1. Sender writes a contract – C• simple – not fully detailed complete• gaps – potentially not obligationally complete
2. Sender observes the state• contract induces action• gap induces communication
3. Communication subgame – ΓC
• sender sends message• receiver takes action
Goal: characterize sender-optimal SPEa
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Introduction Model Example Results Extensions Conclusions
Contract writing game G (K , b)
• Sender writes contract C = {(Ck , xk)}Kk=1
• Clauses (Ck , xk), k = 1, . . . ,K
• Conditions are intervals in the state space, Ck ⊆ [0, 1]
• Instructions xk ∈ R
• K ≤ K
• Commitment:If θ ∈ Ck , action xk implemented
θ
0 1C1 C2
x1 x2
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Introduction Model Example Results Extensions Conclusions
Communication subgame ΓC
• Gap in the contract: L(C) := [0, 1] \⋃Kk=1 Ck
• Sender strategy (messages) σ : L(C)→ ∆(M)
• Receiver strategy (actions) ρ : M → R
• No commitment
• Partitional equilibria• critical types θi• “steps” = induced actions yi
• Γ0 induced by C0 is a CS game:
θ
0 1
y1 y2 y3
m1 m2 m3
θ1 θ2
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Introduction Model Example Results Extensions Conclusions
Communication subgame ΓC
• Gap in the contract: L(C) := [0, 1] \⋃Kk=1 Ck
• Sender strategy (messages) σ : L(C)→ ∆(M)
• Receiver strategy (actions) ρ : M → R
• No commitment
• Partitional equilibria• critical types θi• “steps” = induced actions yi
θ
0 1
y1 y2 y3 y4
m1 m2 m3 m4
θ1 θ2 θ3
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Introduction Model Example Results Extensions Conclusions
Example
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Introduction Model Example Results Extensions Conclusions
Example: K = 1 and b = 13
• Uniform distribution, quadratic payoffs, constant bias• Sender: US (y , θ, b) = −(θ + b− y)2
• Receiver: UR (y , θ) = −(θ − y)2
• There cannot be an equilibrium with more than two steps
• We compare:no contract, 0-step, 1-step, and 2-step optimal contracts
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Introduction Model Example Results Extensions Conclusions
Example: no contract, 0-step
No contract = CS communication:
C∗0 :θ
0 1
y
Obligationally complete contract = no communication (0-step):
C∗0 :θ
0 1
x
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Introduction Model Example Results Extensions Conclusions
Example: 1-step and 2-step
Allowing for 1-step communication:
C∗1 :θ
0 1
xy
Allowing for 2-step communication:
C∗2 :θ
0 1
x
θ1
y1 y2
The sender’s payoffs are ordered:no contract ≺ obl. complete contract ≺ 1-step ≺ 2-step contract
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Introduction Model Example Results Extensions Conclusions
Example with different parameters
Increase number of clauses to K = 2, keeping b = 13 :
C∗0 :θ
0 1
x1 x2
Increase number of clauses to K = 2 and decrease bias to b = 15
(recall: without contract maximally two actions in equilibrium):
C∗3 :θ
0 1
x1 x2y1 y2 y3
θ1 θ2
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Introduction Model Example Results Extensions Conclusions
Example: take-aways
• More clauses improve payoff
• More clauses can drive out communication
• Communication can replace contracting for smaller bias
• More communication actions with contract compared to CS:
Contract relaxes incentive constraints in communication
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Introduction Model Example Results Extensions Conclusions
General results
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Introduction Model Example Results Extensions Conclusions
Maximal use of clausesThe sender optimally uses as many clauses as possible:
Proposition 1
If C = {(Ck , xk)}Kk=1 is an optimal contract in G (K , b), then
K = K .
• Intuition:• Replace communication interval: sender imposes her bias• Split existing clause: actions more precise
θ
0 1
y1 y2 y3θ1 θ2
x
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Introduction Model Example Results Extensions Conclusions
Maximal use of clausesThe sender optimally uses as many clauses as possible:
Proposition 1
If C = {(Ck , xk)}Kk=1 is an optimal contract in G (K , b), then
K = K .
• Intuition:• Replace communication interval: sender imposes her bias• Split existing clause: actions more precise
θ
0 1
y1 y2 y3θ1 θ2
x
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Introduction Model Example Results Extensions Conclusions
Many clauses
If the maximal number of clauses goes to infinity, contractingdrives out communication:
Proposition 2
For any sequence of {LK}∞K=1
of gaps arising in sender-optimal
equilibria e(K , b) of contract-writing games G (K , b),K = 1, 2, . . . ,
limK→∞
Prob(LK ) = 0.
θ0 1
. . . xk . . .
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Introduction Model Example Results Extensions Conclusions
Decreasing bias
If the bias goes to zero, communication drives out contracting:
Proposition 3
Suppose that the continuity property holds for the games Γ0(bi ).For any sequence {Li}∞
i=1 of gaps in sender-optimal equilibria
e(bi ) of games G (K , bi ) with limi→∞ bi = 0,
limi→∞
Prob(Li ) = 1.
θ0 1
x3x1 x2
. . . yn . . .
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Introduction Model Example Results Extensions Conclusions
Results for uniform-quadraticenvironment
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Introduction Model Example Results Extensions Conclusions
Results for uniform-quadratic environment
(Not necessarily optimal) contracts can increase the number ofsteps in communication:
Proposition 4
For any b, there exist a K and a contract C such that there is anequilibrium of the communication subgame ΓC with n inducedactions if and only if n < 1 + 1
2b .
• Comparison to CS for b < 12 :
1
2+
√1
4+
1
2b< 1 +
1
2b
• Example b = 110 : NCS = 2 and N = 5
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Introduction Model Example Results Extensions Conclusions
Many clauses: no communication
Sufficiently many clauses – relative to the bias – result in nocommunication:
Proposition 5
If K > 12b , then any optimal contract will cover [0, 1].
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Introduction Model Example Results Extensions Conclusions
Contracts versus communicationAn optimal contract relaxes incentive constraints:
• Every “condition cluster” contains a critical type(equivalently, no condition cluster belongs to the interior of acommunication interval)
• For meaningful communication, there is a condition clusterwith a critical type that is not 0 or 1
Proposition 6
Suppose that the contract C = {(Ck , xk)}Kk=1 is optimal in thecontract-writing game G , and the equilibrium eC is sender-optimalin the communication subgame ΓC . Then, for every conditioncluster C , there is a critical type θ with C ∩ {θ} 6= ∅. If, inaddition, the equilibrium eC induces at least two communicationactions, then there is a condition cluster C and a critical typeθ 6= 0, 1 with C ∩ {θ} 6= ∅.
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Introduction Model Example Results Extensions Conclusions
No condition in interior
Intuition:
θ0 1θ1 θ2
y1 y2 y3
θ0 1θ1 θ2
y1 y2 y3
θ0 1θ1 θ2
y1 y2 y3
?
θ0 1θ1 θ2
y1 y2 y3
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Introduction Model Example Results Extensions Conclusions
Condition has interior type
Intuition:
C:θ
0 1
x
θ1 θ2
y1 y2 y3
C ′:θ
0 1
x ′
θ1 θ2
y ′1 y2 y3
Cλ:θ
0 1
xλ
θλ1 θλ
2
yλ1 yλ
2 yλ3
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Introduction Model Example Results Extensions Conclusions
Structure of optimal contracts
• Equilibrium is partitional and monotonic
Corollary 7
Suppose that the contract C = {(Ck , xk)}Kk=1 is part of asender-optimal equilibrium eG in the contract-writing game G andinduces a sender-optimal n-step equilibrium eC in thecommunication subgame ΓC . Then, the equilibrium eG is
1. partitional – there is a partition P ={P1,P2, . . . ,PK+n
}of
the type space [0, 1] into intervals such that each P ∈ P iseither a condition of C or a communication interval in eC ; and,
2. monotonic – for any two P,P ′ ∈ P , P 6= P ′, withinf(P ′) ≥ sup(P), the actions a(P ′) and a(P) taken forstates in P ′ and P satisfy a(P ′) > a(P).
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Introduction Model Example Results Extensions Conclusions
Structure of optimal contracts
• Equilibrium is partitional and monotonic
θ
x , y
0 1C1 C2 C3
x1
x2
x3
θ1 θ2 θ3
y1y2
y3
y4
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Introduction Model Example Results Extensions Conclusions
Equalizing communication intervals
• Contracts relax incentive constraints
• Lengths of communication intervals can be equalized
Corollary 8
Suppose K = 1, the contract C with condition [C ,C ] is optimal,and C induces at least two communication actions in thesender-optimal equilibrium eC of the communication subgame ΓC .If θi−1, θi , and θi+1 are critical types in the equilibrium eC withθi ∈
[C ,C
], then
∣∣θi+1 − C∣∣ < |C − θi−1|+ 4b; and, if
θi ∈(C ,C
), then
∣∣θi+1 − C∣∣ ≤ |C − θi−1|.
C Cθi−1 θi θi+1
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Introduction Model Example Results Extensions Conclusions
Extensions
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Introduction Model Example Results Extensions Conclusions
Finite unions of disjoint closed intervals as conditions
• It is never optimal to split the condition into finitely manydisjoint intervals.
Proposition 9
Suppose that we allow contracts with conditions C that are finiteunions of disjoint closed intervals. Then, for b > 1
4 and K = 1,any optimal contract is nonempty and the condition in thatcontract is a single interval.
C:θ
0 1θ1D1 Dj ′ . . .Dj ′′ DJ
y1 y2
C:θ
0 1θ1C C
y1 y2
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Introduction Model Example Results Extensions Conclusions
Example: nonconstant bias
• Assume b(θ) = 13 +
130θ
• Optimal contract covers states with relatively higher bias
θ0 1
x
xb
θ0 1
x
xb
θ0 1
x
xb
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Introduction Model Example Results Extensions Conclusions
Example: nonuniform distribution
• Assume f (θ) = 910 +
210θ
• Optimal contract covers more likely states
θ0 1
x
xf
θ0 1
x
xf
θ0 1
x
xf
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Introduction Model Example Results Extensions Conclusions
Example: transfers• Sender: US (y , θ, b,w) = −(θ + b− y)2−w• Receiver: UR(y , θ,w) = −α(θ − y)2 + (1− α)w
• Sender maximizes:EUS (y , θ, b,w) s.t. EUR(y , θ,w) = uR
• Optimal contract covers fewer states
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Introduction Model Example Results Extensions Conclusions
Extensions: take-aways
• Some motivation for assuming intervals
• Some robustness with respect to: bias, distribution, transfers
• Contract covers states with higher conflict
• Contract covers states that are more likely
• Transfers reduce the set of states in the contract
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Introduction Model Example Results Extensions Conclusions
Conclusions
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Introduction Model Example Results Extensions Conclusions
Concluding remarks
• Model of interaction between contracts and communication
• Tradeoff: ex ante commitment versus ex post discretion
• At the extremes:• Small disagreement: communication dominates• Many clauses: contracting dominates
• Insight: two benefits from contracts• direct: shift control to principal• indirect: relaxation of incentive constraints→ potential for more actions induced by communication→ potential for more equalized communication intervals
• Equilibria are partitional and monotonic
• Optimal contracts• cover states that have more conflict• cover states that are more likely• with transfers cover smaller sets of states
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Introduction Model Example Results Extensions Conclusions
Thank you!
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Introduction Model Example Results Extensions Conclusions
Continuity assumption
Nothing unexpected happens for b → 0:
Continuity Property. For any sequence of biases {bi}∞i=1 with
bi → 0 and any sequence {e(bi )}∞i=1 of sender-optimal equilibria
in the games {Γ0(bi )}∞i=1, the sender’s payoffs in those equilibria
converge to∫[0,1] U
S (yS (θ), θ, 0)dF (θ).
back
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